The quote provided above doesn't indicate that the horizontal stab is producing positive lift, just that the downward force is reduced, along with trim drag.
FBW means that more options exist for creating stability beyond neutral point issues and the positive stability created by the use of a downloaded horizontal stab. An airplane requiring extreme maneuverability can operate with an unloaded horizontal stab and use computer input to create an artificial stability of any form desired, for use by the pilot.
In the case of the A380 described in the quoted paragraph above, the fly by wire technology has been employed to reduce drag and the download by replacing traditional stability with artificial compensation...but it still uses a download...it's just reduced.